The "conservative" of Conservative Judaism is not the "conservative" of Christian/Muslim/Right Wing/etc. conservatisms.

The Conservative Movement began as a middle ground moderation between the ultra orthodoxy and the very liberal Reform Movement in I think it was the 1800s of American Jewish life.

My Conservative temple wherein I was schooled and attended services for five years leading up to Bar Mitzvah was in part funded by Buddy Hackett, not the most conservative guy to come to mind. We had one of the best gymnasiums in Jersey with trampoline I used to play on, (professional basketball teams sometimes practiced there) and we had an indoor swimming pool where I did laps and a steam and sauna where I cruised just below the sanctuary. Not exactly Orthodoxy.

In Judaism, simply put, it is the ultra-Orthodox and the Orthodox who are the conservatives. The Conservatives are the moderates or moderate to liberal. And the Reformed who are liberal to ultra liberal.

...On December 6, 2006, the Committee on Jewish Law and Standards adopted diametrically opposed responsa on the issue of sexual orientation. The CJLS's action permits each congregational rabbi and rabbinical school to decide which responsum to adopt and hence set its own policy on the subject. The adoption of dual, contradictictory responsa represents a straddling of the contemporary societal divide over sexual matters. It also represents a sharp change from previous Conservative policy, which in 1993 had adopted a consensus position reaffirming a blanket prohibition on homosexual conduct while welcoming gay, bisexual, and lesbian people as members....

...On March 26, 2007, the Jewish Theological Seminary in New York followed suit and began accepting openly gay, bisexual, and lesbian candidates for admission for their Rabbinical program...

(further, regarding even the more conservative of the Conservative movement, mostly outside of the USA: )

The Masorti Movement's Israeli Seminary also rejected a change in its view of the status of homosexual conduct, stating that "Jewish law has traditionally prohibited homosexuality." [19] However, the Seminary reversed its position in April 2012, when it approved the ordination of gay, bisexual, and lesbian rabbis in Israel...

(further)

...On June 2012, the American branch of Conservative Judaism formally approved same-sex marriage ceremonies in a 13-0 vote...

So this ought not be a big deal for the rabbi's Conservative congregation, no more so than it would be for any of us to come out, come out, wherever we are.